Everything you have just described IS a budget.
The complaints aimed at Levy are usually that we take the cheap options or focus on resale value ahead of footballing success. So I'm saying, set the budget, be involved in negotiations from a skill point of view, but the DoF decides the sign/walk away point. If the scouting and couching team decide it's best to spend a bit more on a player, do it. They get judged on results on the pitch, give them the tools they ask for, within the parameters set out.
"Want two players at £20m package each? Well, players A is £26m as a package. If you want him, you have to accept you can't have the other one - you decide what will bring success."
If Levy can then negotiate to £22m - happy days!
Maybe the coaching team say - "ok, let's do it. But pull the plug around £24m, we need the extra for player B".
Levy becomes SFO and chief negotiator - things he is great at.
DoF has responsibility for creating the sporting success from the budget, and is measured against it as such.